Structure of the optimal path to a fluctuation

Macroscopic fluctuations have become an essential tool to understand physics far from equilibrium due to the link between their statistics and nonequilibrium ensembles. The optimal path leading to a fluctuation encodes key information on this problem, shedding light on, e.g., the physics behind the enhanced probability of rare events out of equilibrium, the possibility of dynamic phase transitions, and new symmetries. This makes the understanding of the properties of these optimal paths a central issue. Here we derive a fundamental relation which strongly constrains the architecture of these optimal paths for general $d$-dimensional nonequilibrium diffusive systems, and implies a nontrivial structure for the dominant current vector fields. Interestingly, this general relation (which encompasses and explains previous results) makes manifest the spatiotemporal nonlocality of the current statistics and the associated optimal trajectories.

Figure 1

Optimal solution for the current vector field (a) and the density field (b) and (c) associated with a given current fluctuation in the $2d$ Kipnis-Marchioro-Presutti model of heat transport in contact with two boundary thermal baths at temperatures $\rho (x=0)={\rho }_0=3$ and $\rho (x=1)={\rho }_1=1$ and no external field (in this model the locally conserved density field is the energy). Gray lines in (a) depict both local components of the optimal current vector field, while red arrows show the resultant vectors. Note the nontrivial structure of the $y$ component of the current field along the gradient $x$ direction, in stark contrast with the constant, structureless current $x$ component.